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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Analytical hierarchy ： ウィキペディア英語版 Analytical hierarchy In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers, $\backslash mathbb$, and over functions from $\backslash mathbb$ to $\backslash mathbb$. The analytical hierarchy of sets classifies sets by the formulas that can be used to define them; it is the lightface version of the projective hierarchy. == The analytical hierarchy of formulas == The notation $\backslash Sigma^1\_0\; =\; \backslash Pi^1\_0\; =\; \backslash Delta^1\_0$ indicates the class of formulas in the language of second-order arithmetic with no set quantifiers. This language does not contain set parameters. The Greek letters here are lightface symbols, which indicate this choice of language. Each corresponding boldface symbol denotes the corresponding class of formulas in the extended language with a parameter for each real; see projective hierarchy for details. A formula in the language of second-order arithmetic is defined to be $\backslash Sigma^1\_$ if it is logically equivalent to a formula of the form $\backslash exists\; X\_1\backslash cdots\; \backslash exists\; X\_k\; \backslash psi$ where $\backslash psi$ is $\backslash Pi^1\_$. A formula is defined to be $\backslash Pi^1\_$ if it is logically equivalent to a formula of the form $\backslash forall\; X\_1\backslash cdots\; \backslash forall\; X\_k\; \backslash psi$ where $\backslash psi$ is $\backslash Sigma^1\_$. This inductive definition defines the classes $\backslash Sigma^1\_n$ and $\backslash Pi^1\_n$ for every natural number $n$. Because every formula has a prenex normal form, every formula in the language of second-order arithmetic is $\backslash Sigma^1\_n$ or $\backslash Pi^1\_n$ for some $n$. Because meaningless quantifiers can be added to any formula, once a formula is given the classification $\backslash Sigma^1\_n$ or $\backslash Pi^1\_n$ for some $n$ it will be given the classifications $\backslash Sigma^1\_m$ and $\backslash Pi^1\_m$ for all $m$ greater than $n$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Analytical hierarchy」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.